ارايه‌ي روشي جديد براي حل عددي مدل توسعه‌يافته‌ي نفت سياه با در نظر گرفتن اثر فشار مويينگي

A HIGH-RESOLUTION NUMERICAL METHOD FOR SOLVING MULTI-PHASE FLOW IN POROUS MEDIA USING BLACKOIL MODEL WITH CAPILLARY PRESSURE

در اين نوشتار تاثير فشار مويينگي بر جريان چندفاز مخازن هيدروكربني بررسي شده است. براي اين منظور، مدل نفت سياهِ توسعه‌يافته با در نظر گرفتن اثرات فشار مويينگي بسط داده شده است. براي استخراج معادلات حاكم از يك فشار عمومي استفاده شده كه مقدار آن در محدوده‌ي مقادير فشارهاي فازيِ موجود است. معادلات به دست آمده شامل يك معادله‌ي فشار سهموي و يك دستگاه معادلات انتقال اجزا سهموي است كه براي حل آنها از روش ضمني ـ صريح استفاده شده است. براي حل معادله‌ي انتقال اجزا از يك روش مركزي دقت‌بالا استفاده شده كه در مقايسه با ساير روش‌هاي عددي موجود داراي محاسبات كم و دقت قابل قبول است. در انتها نيز سه مسيله‌ي يك‌بعدي نفت سياه كه نتايج آنها در مقالات معتبر علمي موجود است، حل شده و دقت روش عددي به كار رفته مورد بررسي قرار گرفته است.

Capillary pressure plays an important role in production from hydrocarbon reservoirs, especially when gravity drainage is the dominating mechanism, and in fractured reservoirs. There are several capillary pressure models that are fairly accurate for practical purposes if tuned properly for certain cases. The effect of capillary pressure in the governing equations adds some complexities to the numerical simulation of such flows. This can lead to acute numerical challenges when applied to more detailed flow models, like the blackoil model. The blackoil model is a three-phase three-component representation of multiphase flows, in which three components (oil, gas and water) coexist in three phases (liquid, vapor and aqua). In this model, water only appears in the aqua phase, while the other two components can be present in both the liquid and vapor phases. These phases, of course, have different pressures when capillary pressure is considered, and defining a proper pressure, based on which is the development of governing equations, is a delicate matter. The final governing equations include an elliptic pressure equation and a set of parabolic convection-dominated convection-diffusion equations. Numerical simulation of the pressure equation is usually a straightforward matter. Solving the component mass conservation equations, however, can be difficult. Most of the numerical methods used for this purpose are Godunov-based methods which need information about the eigen-structure of the equations. This makes the numerical algorithms complex and highly time consuming. This paper presents a high-resolution central scheme for solving blackoil equations with capillary pressure. The vector-based blackoil formulation used in this work was first introduced by Trangenstein and Bell. The numerical scheme used for solving the mass conservation equations is an extension of the work of Kurganov and Tadmor. This provides a numerical scheme that is computationally efficient and its accuracy is independent from the time step size. To assess the performance of the proposed numerical algorithm, a number of one-dimensional benchmark problems are solved. Numerical results are compared with available data in the literature.